""" Verification of Theorem 4.5.1 (Renewal Capture) and related results (Ch. IV). Renewal Libertarianism, Part 1, Chapter IV, Modules 5 and 6. Theorem 4.5.1 states: under the three conditions (i) substrate payoff convexity in response intensity, (ii) escalation-monotone authority gain after defeat, (iii) non-acquiescent optimal response, and with sufficient defeat capacity, substrate's expected post-renewal authority on the targeted dimension strictly exceeds its pre-renewal authority: E[A_d^{post}] > A_d^{pre}. Verification approach: implement substrate's response-choice game numerically with concrete payoff structures that satisfy the three conditions, compute the substrate's optimal response r*, and confirm the theorem's prediction in the high-defeat regime. Confirm separately that the theorem can fail in the low-defeat regime (the "sufficient defeat capacity" condition is real, not trivial). Then verify the comparative statics (Section 4.5.2), the renewal- provocation result (Section 4.6), and the dimensional asymmetry (Section 4.6.3). Run: uv run python renewal_capture.py """ import numpy as np from _helpers import banner def expected_authority(p_defeat, A_defeat, A_success): """E[A^post | response r] = p_defeat * A^defeat(r) + (1 - p_defeat) * A^success(r)""" return p_defeat * A_defeat + (1 - p_defeat) * A_success def evaluate_response_game(responses, A_pre): """Compute net expected authority for each response option and return the optimal response together with a structured table of intermediate values. """ rows = [] for name, params in responses.items(): p = params["p"] cost = params["cost"] if name == "Acquiescent": # Substrate accepts; no defeat outcome. A^post is the success value # for the substrate (i.e., what it retains after acquiescing). E_A = params["A_success"] else: E_A = expected_authority(p, params["A_defeat"], params["A_success"]) net = E_A - cost rows.append({ "name": name, "p": p, "A_defeat": params.get("A_defeat"), "A_success": params["A_success"], "cost": cost, "E_A": E_A, "net": net, }) best = max(rows, key=lambda r: r["net"]) return best, rows def print_response_table(rows, header): print(f"\n {header}") print(" " + "-" * 80) print(f" {'Response':<14} {'p_def':<8} {'A^defeat':<10} {'A^succ':<10} " f"{'cost':<8} {'E[A]':<10} {'Net':<10}") print(" " + "-" * 80) for r in rows: A_def_str = f"{r['A_defeat']:.3f}" if r["A_defeat"] is not None else " - " print(f" {r['name']:<14} {r['p']:<8.3f} {A_def_str:<10} " f"{r['A_success']:<10.3f} {r['cost']:<8.3f} {r['E_A']:<10.4f} {r['net']:<10.4f}") print(" " + "-" * 80) def main(): banner("Chapter IV Renewal Capture Theorem - numerical verification") # Pre-renewal authority on the targeted dimension (normalize to 1.0) A_pre = 1.0 # ----------------------------------------------------------------- # Build a response game that satisfies the three theorem conditions # and has sufficient defeat capacity (high-defeat regime). # ----------------------------------------------------------------- responses_high_defeat = { # Acquiescent: substrate concedes. A^success here is the reduced # authority substrate retains after acquiescing. "Acquiescent": {"p": 0.0, "A_defeat": None, "A_success": 0.60, "cost": 0.00}, # Mild: partial concessions while preserving most authority. # Modest defeat probability, modest gain on defeat. "Mild": {"p": 0.40, "A_defeat": 1.10, "A_success": 0.70, "cost": 0.05}, # Escalatory: political contestation. Higher defeat probability. "Escalatory": {"p": 0.70, "A_defeat": 1.40, "A_success": 0.40, "cost": 0.15}, # Violent: force suppression. Highest defeat probability, highest # consolidation gain on defeat, catastrophic loss on success. "Violent": {"p": 0.90, "A_defeat": 1.90, "A_success": 0.10, "cost": 0.40}, } best_hi, rows_hi = evaluate_response_game(responses_high_defeat, A_pre) print_response_table(rows_hi, "High-defeat regime") print(f"\n Optimal response r* = {best_hi['name']} (net payoff {best_hi['net']:.4f})") # ----------------------------------------------------------------- # Verify the three theorem conditions on the high-defeat regime # ----------------------------------------------------------------- banner("Theorem 4.5.1: verify conditions (i)-(iii) and the conclusion") # (iii) Non-acquiescent optimal response assert best_hi["name"] != "Acquiescent", ( "Optimal response should not be acquiescent in high-defeat regime") print(f"\n (iii) Optimal response is non-acquiescent: r* = {best_hi['name']} [pass]") # (ii) Escalation-monotone authority gain: A^defeat strictly increases with intensity intensity_order = ["Mild", "Escalatory", "Violent"] A_defeats = [responses_high_defeat[r]["A_defeat"] for r in intensity_order] assert all(A_defeats[i] < A_defeats[i + 1] for i in range(len(A_defeats) - 1)), ( "A^defeat should be strictly increasing in response intensity") print(f" (ii) A^defeat strictly increasing in intensity: " f"M={A_defeats[0]} < E={A_defeats[1]} < V={A_defeats[2]} [pass]") # (i) Payoff convexity in response intensity: # A^defeat at intermediate intensity should not exceed the linear # interpolation between extremes (i.e., the function does not bow up # more on the inside than at the endpoints). linear_midpoint = (A_defeats[0] + A_defeats[2]) / 2 assert A_defeats[1] <= linear_midpoint + 1e-10, ( f"A^defeat should be convex in intensity; got {A_defeats[1]} > " f"linear midpoint {linear_midpoint}") print(f" (i) Convexity: A^defeat(E)={A_defeats[1]} <= " f"(A^defeat(M)+A^defeat(V))/2={linear_midpoint} [pass]") # Theorem's conclusion: E[A^post | r*] > A^pre E_post_optimal = best_hi["E_A"] print(f"\n Conclusion: E[A^post | r* = {best_hi['name']}] = {E_post_optimal:.4f}, " f"A^pre = {A_pre:.4f}") print(f" Difference E[A^post] - A^pre = {E_post_optimal - A_pre:+.4f}") assert E_post_optimal > A_pre, ( "Theorem 4.5.1 conclusion E[A^post] > A^pre failed in high-defeat regime") print(f" [pass] E[A^post] > A^pre under the three theorem conditions") # ----------------------------------------------------------------- # Boundary: the conclusion can fail when defeat capacity is insufficient # ----------------------------------------------------------------- banner("Theorem 4.5.1 boundary: conclusion fails when defeat capacity is too low") # Same payoff structure but lower defeat probabilities throughout responses_low_defeat = { "Acquiescent": {"p": 0.00, "A_defeat": None, "A_success": 0.60, "cost": 0.00}, "Mild": {"p": 0.10, "A_defeat": 1.10, "A_success": 0.70, "cost": 0.05}, "Escalatory": {"p": 0.20, "A_defeat": 1.40, "A_success": 0.40, "cost": 0.15}, "Violent": {"p": 0.30, "A_defeat": 1.90, "A_success": 0.10, "cost": 0.40}, } best_lo, rows_lo = evaluate_response_game(responses_low_defeat, A_pre) print_response_table(rows_lo, "Low-defeat regime") print(f"\n Optimal response r* = {best_lo['name']} (net payoff {best_lo['net']:.4f})") print(f" E[A^post | r* = {best_lo['name']}] = {best_lo['E_A']:.4f}, A^pre = {A_pre:.4f}") print(f" Difference E[A^post] - A^pre = {best_lo['E_A'] - A_pre:+.4f}") assert best_lo["E_A"] < A_pre, ( "In low-defeat regime, optimal E[A^post] should fall below A^pre, " "showing the 'sufficient defeat capacity' condition is non-trivial") print(f" [pass] Theorem conclusion fails when defeat capacity is insufficient") print(f" (the 'sufficient defeat capacity' condition is real, not trivial)") # ----------------------------------------------------------------- # Comparative statics (Section 4.5.2) # Higher defeat probability strengthens renewal capture # ----------------------------------------------------------------- banner("Comparative statics (Section 4.5.2): E[A^post] - A^pre is increasing in p") # Sweep defeat probability for the Escalatory response holding everything else fixed A_defeat_fixed = 1.40 A_success_fixed = 0.40 cost_fixed = 0.15 print(f"\n Escalatory response, A^defeat={A_defeat_fixed}, A^success={A_success_fixed}, cost={cost_fixed}") print(f" {'p_defeat':<12} {'E[A^post]':<12} {'Net':<12} {'E[A^post] - A^pre':<20}") print(" " + "-" * 56) prev_diff = -float("inf") for p in np.linspace(0.1, 0.95, 18): E_A = expected_authority(p, A_defeat_fixed, A_success_fixed) net = E_A - cost_fixed diff = E_A - A_pre print(f" {p:<12.4f} {E_A:<12.4f} {net:<12.4f} {diff:<+20.4f}") assert diff > prev_diff, "E[A^post] - A^pre should be monotonically increasing in p" prev_diff = diff print(f"\n [pass] E[A^post] - A^pre strictly increasing in defeat probability p") # Identify the threshold p above which renewal capture occurs # Solve E[A^post] = A^pre: p * A^defeat + (1-p) * A^success = A^pre # p * (A^defeat - A^success) = A^pre - A^success # p = (A^pre - A^success) / (A^defeat - A^success) p_threshold = (A_pre - A_success_fixed) / (A_defeat_fixed - A_success_fixed) print(f"\n Renewal-capture threshold p* on Escalatory response:") print(f" p* = (A^pre - A^success) / (A^defeat - A^success) = {p_threshold:.4f}") # Verify: E[A^post] crosses A^pre exactly at p_threshold E_at_thresh = expected_authority(p_threshold, A_defeat_fixed, A_success_fixed) assert abs(E_at_thresh - A_pre) < 1e-12, "Threshold formula does not produce E[A^post] = A^pre" print(f" [pass] E[A^post](p=p*) = {E_at_thresh:.6f} = A^pre (threshold formula verified)") # ----------------------------------------------------------------- # Section 4.6: Renewal Provocation result # Substrate has positive marginal incentive to provoke renewal attempts # when E[A^post] - A^pre > 0 (which is the high-defeat regime's # conclusion of Theorem 4.5.1). # ----------------------------------------------------------------- banner("Renewal Provocation (Section 4.6): incentive to provoke when defeat capacity is high") provocation_payoff = best_hi["E_A"] - A_pre print(f"\n In the high-defeat regime, substrate's optimal r* gives:") print(f" E[A^post] - A^pre = {provocation_payoff:+.4f}") print(f" This positive expected gain is substrate's incentive to provoke the renewal") print(f" attempt, since the attempted renewal raises substrate's expected authority") print(f" above the pre-renewal level.") assert provocation_payoff > 0, "Provocation incentive should be positive in high-defeat regime" print(f" [pass] positive marginal incentive to provoke confirmed (E[A^post] > A^pre)") print(f"\n In the low-defeat regime, substrate's optimal r* gives:") print(f" E[A^post] - A^pre = {best_lo['E_A'] - A_pre:+.4f}") print(f" The expected outcome is below A^pre, so substrate has no provocation") print(f" incentive in this regime; the renewal attempt would lower substrate") print(f" authority in expectation.") assert best_lo["E_A"] - A_pre < 0 print(f" [pass] no provocation incentive in low-defeat regime") # ----------------------------------------------------------------- # Section 4.6.3: Dimensional asymmetry # The same polity can have substrate choosing different responses on # different dimensions, if defeat capacity differs across dimensions. # ----------------------------------------------------------------- banner("Dimensional asymmetry (Section 4.6.3): different optimal responses across dimensions") # Suppose substrate has high defeat capacity on extraction (tau) but low # on scope (S) -- a polity with strong central authority over revenue but # weak central authority over jurisdictional boundaries. print(f"\n Polity with high defeat capacity on tau, low on S, both targeted simultaneously.") best_tau, _ = evaluate_response_game(responses_high_defeat, A_pre) best_S, _ = evaluate_response_game(responses_low_defeat, A_pre) print(f" Optimal response on tau-targeting attempt: {best_tau['name']} " f"(E[A^post]={best_tau['E_A']:.4f})") print(f" Optimal response on S-targeting attempt: {best_S['name']} " f"(E[A^post]={best_S['E_A']:.4f})") assert best_tau["name"] != best_S["name"], ( "Substrate should choose different responses across dimensions when defeat capacities differ") print(f" [pass] substrate rationally picks different responses across dimensions") print(f" (escalation on tau where defeat capacity is high; less-costly") print(f" response on S where defeat capacity is low)") banner("All verifications passed") print(""" Result: Chapter IV's Renewal Capture Theorem and adjacent results verified on concrete numerical instances of substrate's response-choice game. - Theorem 4.5.1: conditions (i) convexity, (ii) escalation-monotone authority gain, (iii) non-acquiescent optimum all hold by construction; conclusion E[A^post] > A^pre verified in the high-defeat regime. - Boundary: in the low-defeat regime, the optimal response still satisfies condition (iii) but the conclusion fails, confirming the 'sufficient defeat capacity' qualifier is non-trivial. - Section 4.5.2 comparative statics: E[A^post] - A^pre is strictly monotonically increasing in defeat probability p. - Renewal-capture threshold formula p* = (A^pre - A^success) / (A^defeat - A^success) verified by direct substitution. - Section 4.6 Renewal Provocation: positive marginal incentive to provoke confirmed in the high-defeat regime; absent in the low-defeat regime. - Section 4.6.3 Dimensional asymmetry: substrate rationally picks different optimal responses across dimensions with different defeat capacities, even in the same polity at the same time. """) if __name__ == "__main__": main()